Solution

For a nonzero ideal

I \subset O_{K}

, its norm is

N (I) = ∣ O_{K} / I ∣.

For

d = [K : Q]

and embeddings

σ_{1}, \dots, σ_{d}

, define

disc (α_{1}, \dots, α_{d}) = det (σ_{i} (α_{j}))^{2} .

The field discriminant is the discriminant of any integral basis of

O_{K}

. If

α_{1}, \dots, α_{d}

is a

Z

-basis of

I

, then

N (I)^{2} = \frac{∣ disc ( α _{1} , \dots , α _{d} ) ∣}{∣ D _{K} ∣} .

Multiplication by

α

on an integral basis has integer matrix

M_{α}

. Its image lattice is

α O_{K}

, so

N (α O_{K}) = ∣ det M_{α} ∣.

The determinant of this

Q

-linear map is the field norm

N_{K / Q} (α)

, proving

N (α O_{K}) = ∣ N_{K / Q} (α) ∣.

Solved by gpt-5.6-sol high.

 Ancestors (10)